hadronic matrix element for four quark condensate?

Question

I am looking at the neutral kaon mixing in SM where calculation of a four quark condensate between Kaon and anti-Kaon is required, in order to obtain the kaon mass difference.

$$\langle K| [ \bar{d}\gamma^\mu(1-\gamma_5)s][\bar{d}\gamma_\mu(1-\gamma_5)s]|\bar{K}\rangle=\frac83 \langle K|\bar{d}\gamma^\mu\gamma_5s|0\rangle\langle 0|\bar{d}\gamma_\mu\gamma_5s|\bar{K}\rangle=\frac83 \frac{f_K^2 m_K^2}{2m_K}.$$

This equation is copied from Page 380 of Li&Cheng.

Can anyone explain to me

1, the factor $(2m_K)^{-1}$ which "arises from the normalization of the state"?

2, the factor $8/3$ where $4$ is the number of Wick contraction ways and $2/3$ is a color factor? I do not understand this color factor here.

3, most importantly, in general how to calculate such four quark condensates, with different handedness combinations of the quarks, and different (pesudoscalar) external "sandwiching" mesons?

I understand the method used in this equation is called Vacuum Saturation Approximation and its main idea. But really unsure about how to determine the prefactor (in this case 8/3) of RHS..

Thank you very much!

Answer 1

1) The $(2m_K)$ comes from the LSZ formula (and depends on your normalization).

2) The 8/3=2*4/3 comes from Wick contractions and Fierz identities. Consider the possible contractions of the 4-quark operator. We are only interested in contractions of the form $[\bar{d}s]$, because only these terms can generate vacuum-to-kaon matrix elements. There is a simple color singlet contraction, and a non-singlet term $$ [\bar{d}_\alpha^a s_\beta^b][\bar{d}_\rho^c s_\sigma^d] \;\; (\delta_{ad}\delta_{bc})\;\; ((\gamma_\mu\gamma_5)_{\alpha\sigma}(\gamma_\mu\gamma_5)_{\rho\beta} +(\gamma_\mu)_{\alpha\sigma}(\gamma_\mu)_{\rho\beta}] $$ I can use Fierz identities to rearrange the index structure. The color singlet, Dirac (V-A)(V-A) term has a coefficient 1/3 (color) * 1 (Dirac). Adding the two contributions gives 1+1/3=4/3. There are two ways of inserting the vacuum state, the first bilinear can either create a kaon or annihilate an anti-kaon. This gives 8/3.

3) Finally, I use the definition of $f_K$ in terms of the axial vector matrix elements of the kaon.